CN107102554A - A kind of magnetic suspension spherical flywheel unbalance vibration suppressing method - Google Patents
A kind of magnetic suspension spherical flywheel unbalance vibration suppressing method Download PDFInfo
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Abstract
The present invention relates to a kind of magnetic suspension spherical flywheel unbalance vibration suppressing method.According to Newton's second law and gyroscope technology establishing equation magnetic bearing rotor dynamics equation, based on D'Alembert's principle, obtain rotor inertia axle deviate unbalance mass, square caused by geometrical axis the rotor centre of form is crossed to the perturbed force and suspending power of rotor to eccentric throw caused by the perturbed force of rotor, rotor centroid offset geometry axle and deflection negative moment that only barycenter is caused to the perturbed force of rotor.Displacement under displacement and magnetic bearing coordinate system that rotor displacement amount under three kinds of perturbed force effects is respectively converted into by transition matrix under sensor coordinate system, and be applied to two kinds of displacements in controller and magnetic bearing respectively, three kinds of perturbed forces are suppressed using feedforward suppressing method.The present invention can effectively improve the control accuracy of magnetic axis bearing rotor system unbalance vibration.
Description
Technical field
The present invention relates to a kind of magnetic suspension spherical flywheel unbalance vibration suppressing method, it is adaptable to which rotor is spherical structure
The unbalance vibration control of magnetic suspension spherical flywheel.
Background technology
Mechanical bearing there are problems that fretting wear and, it is impossible to meet spacecraft attitude control system high
Precision and long-life demand.Magnetically levitated flywheel carries out contactless suspension bearing using magnetic bearing, eliminates rubbing for mechanical bearing
Scouring is damaged, and with Vibration Active Control and suppression function, disclosure satisfy that the need of spacecraft attitude control system high precision and long service life
Ask.Under existing magnetically levitated flywheel pole air-gaps are post shelly, conical shell shape or thin wall shape, rotor deflection state, pole air-gaps shape
Shape changes, and magnetic is close uneven in air gap, causes rotor surface magnetic force to be distributed by certain gradient, produces deflection negative moment, reduces
The suspension precision and control accuracy of magnetically levitated flywheel.Apply for a patent a kind of 201510813055.9 internal rotor magnetic suspension sphere gyros
Flywheel proposes a kind of magnetically suspended gyroscope flywheel of spherical spinner structure, and flywheel uses spherical structure rotor, and passes through sphere magnetic
Pole magnetic bearing supporting, can be prevented effectively from the air gap deformation problems of aspherical rotor presence, overcome magnetic to draw inclined negative moment, improve
The suspension precision and control accuracy of magnetically suspended gyroscope flywheel.
Due to the reason such as rotor actual processing assembly precision and quality of materials uniformity, there is certain imbalance in rotor
Amount, when rotating at a high speed, amount of unbalance can cause rotor larger vibration, reduce rotor suspension precision.For rotor unbalance vibration,
The interference source for causing rotor oscillation is analyzed, corresponding magnetic bearing-dynamical model of rotor is set up, and obtain suffered by rotor
The expression formula of perturbed force, is suppressed by corresponding control method to rotor perturbed force, so as to reach suppression rotor unbalance
The effect of vibration.Rotor interference source is analyzed, and sets up corresponding magnetic bearing-dynamical model of rotor, it is uneven as rotor is suppressed
Weigh the committed step vibrated, significant to suppressing rotor unbalance vibration.Existing magnetically levitated flywheel unbalance vibration suppression
In method processed, a few methods only consider the rotor centroid caused interference of eccentric throw to rotor not on rotor geometrical axis, and build
Found corresponding magnetic bearing-dynamical model of rotor.And most methods are to ignore eccentricity of rotor, rotor inertia axle is only considered
Interference of the unbalance mass, square to rotor caused by deviateing geometrical axis, sets up magnetic bearing-dynamical model of rotor, and basic herein
It is upper that rotor oscillation is suppressed using corresponding control method.Magnetic suspension spherical flywheel is spherical spinner structure, by rotor machining precision
Limitation, rotor centroid deviates the rotor centre of sphere, causes rotor to there is eccentric throw, because footpath/axial magnetic bearing has leakage field, deflects magnetic
Magnetic is close uneven at bearing edge, if ignoring rotor eccentricity square, rotor does not rotate around geometrical axis, and rotor can be interfered the shadow of power
Ring, while the uniformity of stator air gap can be reduced, reduce magnetic suspension spherical flywheel control accuracy, therefore eccentric throw be can not ignore.
Further, since rotor sole mass distributing homogeneity can not ensure that rotor has unbalance mass, rotor inertia axle is caused to deviate
Geometrical axis, causes unbalance mass, square, causes rotor unbalance to vibrate, therefore, to consider that rotor centroid deviates rotor ball simultaneously
Eccentricity caused by the heart and rotor inertia axle deviate interference of the unbalance mass, square to rotor caused by geometrical axis.Simultaneously as
The presence of eccentric throw, suspending power crosses rotor centroid and only the centre of sphere causes to deflect negative moment, disturbs rotor motion, reduces flywheel
Control accuracy and suspension precision, so, consider eccentricity, unbalance mass, square and deflection three kinds of interference sources of negative moment,
Corresponding magnetic bearing-dynamical model of rotor is set up, rotor unbalance vibration is suppressed, magnetic suspension spherical flywheel is improved
Control accuracy and suspension precision.
The content of the invention
The technology of the present invention solves problem：For the magnetic suspension spherical flywheel unbalance vibration with spherical spinner, carry
A kind of magnetic suspension spherical flywheel unbalance vibration suppressing method is gone out, the rotor displacement amount under perturbed force is acted on is by changing square
Battle array is respectively converted into the displacement under the displacement under sensor coordinate system and magnetic bearing coordinate system, and two kinds of displacements are distinguished
It is applied in controller and magnetic bearing, three kinds of perturbed forces is suppressed using feedforward suppressing method.
The present invention technical solution be：According to Newton's second law and gyroscope technology establishing equation magnetic bearing-rotor
Kinetics equation, based on D'Alembert's principle, obtains rotor inertia axle and deviates unbalance mass, square caused by geometrical axis to rotor
Eccentric throw caused by perturbed force, rotor centroid offset geometry axle the rotor centre of form is crossed to the perturbed force and suspending power of rotor and only
Perturbed force of the deflection negative moment that barycenter is caused to rotor.Rotor displacement amount under three kinds of perturbed force effects is passed through into transition matrix
Displacement under the displacement and magnetic bearing coordinate system that are respectively converted under sensor coordinate system, and two kinds of displacements are made respectively
Use in controller and magnetic bearing, three kinds of perturbed forces are suppressed using feedforward suppressing method, following steps are specifically included：
1st, according to Newton's second law and gyroscope technology establishing equation magnetic suspension spherical flywheel magnetic bearing-rotor dynamics side
Cheng Wei：
Abbreviation is obtained
By the electromagnetism of each suspension passage of rotor it is force linearizing after obtain：
Abbreviation obtains Fm=Khqm+KiI
Wherein,
M=[m Jy m JxM], q=[x β y-α z]T、
F=[fx py fy -px fz]TRespectively Mass matrix, gyro battle array, generalized coordinates and generalized force；Fm=[fx py fy px
fz]TThe electromagnetic force and torque acted on for magnetic bearing on rotor；Kh=diag [khx 0khy 0khz] it is magnetic bearing displacement rigidity
Battle array, khx、khy、khzDisplacement rigidity of the rotor along X, Y, Z axis translation is represented respectively；qm=[xm 0 ym 0 zm]TFor magnetic bearing coordinate
It is lower rotor part displacement, xm、ym、zmDisplacement of the magnetic bearing coordinate system lower rotor part in X, Y, Z axis direction is represented respectively；
Ki=diag [kix kiβ kiy kiα kiz] it is magnetic bearing current stiffness battle array, kix、kiy、kizMagnetic bearings control is represented respectively
Current stiffness of the rotor along X, Y, Z axis translation, kiα、kiβRepresent that magnetic bearings control rotor is firm around the electric current that X, Y-axis are deflected respectively
Degree；I=[ix iβ iy iα iz]TFor magnetic bearings control electric current battle array, ix、iy、izRepresent control rotor along X, Y, Z axis translation respectively
Electric current, iα、iβThe electric current that control rotor is deflected around X, Y-axis is represented respectively；The quality of rotor is m, position of the rotor along X, Y and Z axis
It is respectively x, y and z to move, and rotor is respectively α, β around X, Y-axis angular displacement, and the angular speed of rotor about the z axis is ω, magnetic bearing effect
In rotor geometric center make a concerted effort and resultant moment be respectively fx、fy、fzAnd Px、Py, rotary inertia point of the rotor around X, Y and Z axis
Wei not Jx、JyAnd Jz。
2nd, based on D'Alembert's principle, obtain rotor inertia axle deviate geometrical axis caused by unbalance mass, square rotor is done
Disturb power.
If apex rotor is A ends, there is unbalance mass, square g in the duplicate removal face of A endsa=maR, wherein maFor in the duplicate removal face of A ends not
Balance mass, r is duplicate removal radius.
Ignore external interference, following five equations can be listed according to D'Alembert's principle：
Wherein Fax、Fay、FazComponent of the perturbed force in X, Y, Z axis suffered by rotor A end, F are represented respectivelygx、FgyRespectively
Rotor inertial forces are represented in X, the component of Y-axis,Respectively represent rotor unbalance moment of mass caused by perturbed force
X, Y-axis component, FlRepresent active force of the deflection magnetic bearing on flywheel rotor, Mgx、MgyRepresent inertia force to being consolidated in respectively
The inertia couple of X-axis and Y-axis in flywheel equatorial plane, R represents that deflection magnetic bearing coil installs radius, lanRepresent rotor A end
Distance of the duplicate removal face to rotor center plane.
3rd, because flywheel rotor is that inertia force is zero, i.e. F on inertia axially symmetric structuregx=Fgy=0, so by injustice
Weighing apparatus moment of mass be to the perturbed force of rotor：
If rotor bottom is B ends, similarly unbalance mass, square in B ends is gb=mbR, wherein mbFor in the duplicate removal face of rotor B ends not
Balance mass, r is duplicate removal radius, then rotor B ends unbalance mass, square is to the perturbed force of rotor：
Wherein, Fbx、Fby、FbzComponent of the perturbed force in X, Y, Z axis suffered by rotor B ends, l are represented respectivelybnRepresent rotor
Distance of the B ends duplicate removal face to rotor center plane.
4th, in order to ensure the uniformity of magnetic suspension spherical flywheel stator air gap, it is considered to caused by rotor centroid offset geometry axle
Eccentric throw, obtain eccentric throw is to the perturbed force of rotor：
Wherein Fux1、Fuy1、Fuz1Represent component of the eccentric throw to the perturbed force of rotor in X, Y, Z axis, emRepresent that flywheel turns
The eccentric throw of son, Φ is the phase of eccentric throw, and ρ is eccentric throw and magnetic bearing central plane angle.
5th, consider that suspending power crosses the rotor centre of form and deflection negative moment that only barycenter is caused, and obtain thus caused interference
Power is：
I.e.：
Wherein Fux2、Fuy2、Fuz2Represent component of the deflection negative moment to the perturbed force of rotor in X, Y, Z axis.
6th, above-mentioned three kinds of perturbed forces are considered, the rotor displacement amount under perturbed force is acted on is turned respectively by transition matrix
Displacement under the displacement and magnetic bearing coordinate system that are changed under sensor coordinate system, and two kinds of displacements are applied to control respectively
In device and magnetic bearing processed, three kinds of perturbed forces are suppressed using feedforward suppressing method.
Rotor-magnetic bearing transition matrix is E, and the vertical range of sensor to XOY plane is la, sensor to XOZ planes
Horizontal range be l, then rotor-sensor transition matrix TsFor：
The numerical value of disturbance torque is obtained by calculating, feedforward compensation is carried out to disturbance torque by controller, realized to rotor
The suppression of unbalance vibration.Condition, which is fully compensated, according to feedforward control to obtain：
In formula, Gf(s)、Gu(s)、Gp(s) unbalance mass, square, eccentric throw and the feedforward for deflecting negative moment are represented respectively
Transfer matrix, Gw(s) power amplifier is represented.
It can be obtained according to above formula：
Above formula belongs to stable state and compensated entirely, is easily realized in control.
The principle of such scheme is：
Magnetic suspension spherical flywheel is used under spherical spinner structure, rotor deflection state, and air gap shape will not change, and keep away
Exempt from the generation that magnetic draws inclined torque, improve the suspension precision and control accuracy of magnetic suspension spherical flywheel.But by rotor machining
Precision is limited, and the rotor centre of form deviates with the rotor centre of sphere, causes rotor to there is eccentric throw, causes rotor unbalance to vibrate.Meanwhile, by
It can not ensure in rotor quality uniformity, cause rotor to there is unbalance mass, produce unbalance mass, square, cause rotor uneven
Weighing apparatus vibration.Further, since rotor has an eccentric throw, suspending power crosses the rotor centre of form and deflection negative moment that only barycenter is caused,
Rotor unbalance can be caused to vibrate.Therefore, to consider eccentric throw, unbalance mass, square and deflection negative moment to do rotor
Disturb, set up corresponding magnetic bearing-dynamical model of rotor.On this basis, the rotor displacement amount under perturbed force is acted on passes through
Transition matrix is respectively converted into the displacement under the displacement under sensor coordinate system and magnetic bearing coordinate system, and by two kinds of displacements
Amount is applied in controller and magnetic bearing respectively, and the rotor unbalance caused using feedforward control to three kinds of perturbed forces, which is vibrated, to be carried out
Suppress.
The advantage of the present invention compared with prior art is：(1) considering that it is inclined that the rotor centre of form deviation rotor centre of sphere is caused
On the basis of the heart is away from unbalance mass, square both interference sources caused with rotor inertia axle offset geometrical axis, increase considers outstanding
Buoyancy crosses the rotor centre of form and deflection negative moment that only barycenter is caused is to the perturbed force of rotor.(2) rotor-magnetic bearing conversion is set up
Matrix, rotor-sensor transition matrix, lower turn of magnetic bearing coordinate system is respectively converted into by the rotor displacement amount under rotor coordinate
Sub- displacement and sensor coordinate system lower rotor part displacement, and two displacements are compensated into magnetic bearing displacement rigidity respectively and are applied to
In system controller.
Brief description of the drawings
Fig. 1 is flow chart of the invention；
Fig. 2 is magnetic bearing-dynamical model of rotor；
Fig. 3 is feedforward control structured flowchart of the present invention.
Description of reference numerals：1 is axial magnetic bearing, and 2 be radial direction magnetic bearing, and 3 be spherical spinner, and 4 be permanent magnet, and 5 be axle
To sensor, 6 be winding, and 7 be rotor geometrical axis, and 8 be rotor inertia axle.
Embodiment
As shown in figure 1, in specific implementation process, specific implementation step of the invention is as follows：
1st, according to Newton's second law and gyroscope technology establishing equation magnetic suspension spherical flywheel magnetic bearing-rotor dynamics side
Cheng Wei：
Abbreviation is obtained
By the electromagnetism of each suspension passage of rotor it is force linearizing after obtain：
Abbreviation obtains Fm=Khqm+KiI
Wherein,
M=[m Jy m JxM], q=[x β y-α z]T、
F=[fx py fy -px fz]TRespectively Mass matrix, gyro battle array, generalized coordinates and generalized force；Fm=[fx py fy px
fz]TThe electromagnetic force and torque acted on for magnetic bearing on rotor；Kh=diag [khx 0 khy 0 khz] it is magnetic bearing displacement rigidity
Battle array, khx、khy、khzDisplacement rigidity of the rotor along X, Y, Z axis translation is represented respectively；qm=[xm 0 ym 0 zm]TFor magnetic bearing coordinate
It is lower rotor part displacement, xm、ym、zmDisplacement of the magnetic bearing coordinate system lower rotor part in X, Y, Z axis direction is represented respectively；Ki=diag [kix
kiβ kiy kiα kiz] it is magnetic bearing current stiffness battle array, kix、kiy、kizRepresent magnetic bearings control rotor along X, Y, Z axis translation respectively
Current stiffness, kiα、kiβThe current stiffness that magnetic bearings control rotor is deflected around X, Y-axis is represented respectively；I=[ix iβ iy iα
iz]TFor magnetic bearings control electric current battle array, ix、iy、izElectric current of the control rotor along X, Y, Z axis translation, i are represented respectivelyα、iβDifference table
Show the electric current that control rotor is deflected around X, Y-axis；As shown in Fig. 2 the quality of rotor is m, displacement difference of the rotor along X, Y and Z axis
For x, y and z, rotor is respectively α, β around X, Y-axis angular displacement, and the angular speed of rotor about the z axis is ω, and magnetic bearing acts on rotor
In geometric center make a concerted effort and resultant moment be respectively fx、fy、fzAnd Px、Py, rotor is respectively J around X, Y and Z axis rotary inertiax、
JyAnd Jz。
2nd, based on D'Alembert's principle, obtain rotor inertia axle deviate geometrical axis caused by unbalance mass, square rotor is done
Disturb power.
If apex rotor is A ends, there is unbalance mass, square g in the duplicate removal face of A endsa=maR, wherein maFor in the duplicate removal face of A ends not
Balance mass, r is duplicate removal radius.
Ignore external interference, following five equations can be listed according to D'Alembert's principle：
Wherein Fax、Fay、FazComponent of the perturbed force in X, Y, Z axis suffered by rotor A end, F are represented respectivelygx、FgyRespectively
Rotor inertial forces are represented in X, the component of Y-axis,Respectively represent rotor unbalance moment of mass caused by perturbed force
X, Y-axis component, FlRepresent active force of the deflection magnetic bearing on flywheel rotor, Mgx、MgyRepresent inertia force to being consolidated in respectively
The inertia couple of X-axis and Y-axis in flywheel equatorial plane, R represents that deflection magnetic bearing coil installs radius, lanRepresent rotor A end
Distance of the duplicate removal face to rotor center plane.
3rd, because flywheel rotor is that inertia force is zero, i.e. F on inertia axially symmetric structuregx=Fgy=0, so by injustice
Weighing apparatus moment of mass be to the perturbed force of rotor：
If rotor bottom is B ends, similarly unbalance mass, square in B ends is gb=mbR, wherein mbFor in the duplicate removal face of rotor B ends not
Balance mass, r is duplicate removal radius, then rotor B ends unbalance mass, square is to the perturbed force of rotor：
Wherein Fbx、Fby、FbzComponent of the perturbed force in X, Y, Z axis suffered by rotor B ends, l are represented respectivelybnRepresent rotor B
Duplicate removal face is held to the distance of rotor center plane.
4th, in order to ensure the uniformity of magnetic suspension spherical flywheel stator air gap, it is considered to caused by rotor centroid offset geometry axle
Eccentric throw, obtain eccentric throw is to the perturbed force of rotor：
Wherein Fux1、Fuy1、Fuz1Represent component of the eccentric throw to the perturbed force of rotor in X, Y, Z axis, emRepresent that flywheel turns
The eccentric throw of son, Φ is the phase of eccentric throw, and ρ is eccentric throw and magnetic bearing central plane angle.
5th, consider that suspending power crosses the rotor centre of form and deflection negative moment that only barycenter is caused, and obtain thus caused interference
Power is：
I.e.：
Wherein Fux2、Fuy2、Fuz2Represent component of the deflection negative moment to the perturbed force of rotor in X, Y, Z axis.
6th, above-mentioned three kinds of perturbed forces are considered, the rotor displacement amount under perturbed force is acted on is turned respectively by transition matrix
Displacement under the displacement and magnetic bearing coordinate system that are changed under sensor coordinate system, and two kinds of displacements are applied to control respectively
In device and magnetic bearing processed, three kinds of perturbed forces are suppressed using feedforward suppressing method.
Rotor-magnetic bearing transition matrix is E, and the vertical range of sensor to XOY plane is la, sensor to XOZ planes
Horizontal range be l, then rotor-sensor transition matrix TsFor：
The numerical value of disturbance torque is obtained by calculating, feedforward compensation is carried out to disturbance torque by controller, realized to rotor
The suppression of unbalance vibration.Condition, which is fully compensated, according to feedforward control to obtain：
In formula, Gf(s)、Gu(s)、Gp(s) unbalance mass, square, eccentric throw and the feedforward for deflecting negative moment are represented respectively
Transfer matrix, Gw(s) power amplifier is represented.
It can be obtained according to above formula：
Above formula belongs to stable state and compensated entirely, is easily realized in control, can obtain disturbance torque effect under rotor control block such as
Shown in Fig. 3, wherein Gc(s) controller, G are represented0(s) rotor, k are representedsRepresent sensor coefficient.
The content not being described in detail in description of the invention belongs to prior art known to professional and technical personnel in the field.
The above embodiment of the present invention is that the limitation present invention is cannot be used for the explanation of scheme, has protection domain suitable with the present invention
Any change in implication and scope, is all considered as being included in the scope of protection of the invention.
Claims (1)
1. a kind of magnetic suspension spherical flywheel unbalance vibration suppressing method, it is characterised in that：According to Newton's second law and gyro
Technology establishing equation magnetic bearing-rotor dynamics equation, based on D'Alembert's principle, obtains rotor inertia axle deviation geometrical axis and causes
Unbalance mass, square to eccentric throw caused by the perturbed force of rotor, rotor centroid offset geometry axle to the perturbed force of rotor and
Suspending power crosses the rotor centre of form and deflection negative moment that only barycenter is caused is to the perturbed force of rotor；By under three kinds of perturbed force effects
Rotor displacement amount is respectively converted into the displacement under the displacement under sensor coordinate system and magnetic bearing coordinate system by transition matrix
Amount, and two kinds of displacements are applied in controller and magnetic bearing respectively, three kinds of perturbed forces are carried out using feedforward suppressing method
Suppress, specifically include following steps：
(1) according to Newton's second law and gyroscope technology establishing equation magnetic suspension spherical flywheel magnetic bearing-rotor dynamics equation
For：
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<mi>f</mi>
<mrow>
<mi>z</mi>
<mo>-</mo>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>h</mi>
<mi>z</mi>
</mrow>
</msub>
<msub>
<mi>z</mi>
<mi>m</mi>
</msub>
<mo>-</mo>
<msub>
<mi>k</mi>
<mrow>
<mi>i</mi>
<mi>z</mi>
</mrow>
</msub>
<msub>
<mi>i</mi>
<mi>z</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
Abbreviation is obtained：
Fm=Khqm+KiI
Wherein,
Mass matrix M, gyro battle array q, generalized coordinates G and generalized force F are respectively：
M=[m Jy m Jxm]；
Q=[x β y-α z]T；
<mrow>
<mi>G</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<msub>
<mi>J</mi>
<mi>z</mi>
</msub>
<mi>&omega;</mi>
<mo>;</mo>
</mrow>
F=[fx py fy -px fz]T；
Electromagnetic force and torque F that magnetic bearing is acted on rotorm=[fx py fy px fz]T；
Magnetic bearing displacement rigidity battle array Kh=diag [khx 0 khy 0 khz]；
khx、khy、khzDisplacement rigidity of the rotor along X, Y, Z axis translation is represented respectively；
qm=[xm 0 ym 0 zm]TFor magnetic bearing coordinate system lower rotor part displacement, xm、ym、zmLower turn of magnetic bearing coordinate system is represented respectively
Displacement of the son in X, Y, Z axis direction；
Ki=diag [kix kiβ kiy kiα kiz] it is magnetic bearing current stiffness battle array,
kix、kiy、kizCurrent stiffness of the magnetic bearings control rotor along X, Y, Z axis translation is represented respectively,
kiα、kiβThe current stiffness that magnetic bearings control rotor is deflected around X, Y-axis is represented respectively；
I=[ix iβ iy iα iz]TFor magnetic bearings control electric current battle array,
ix、iy、izElectric current of the control rotor along X, Y, Z axis translation is represented respectively,
iα、iβThe electric current that control rotor is deflected around X, Y-axis is represented respectively；
The quality of rotor is m, and displacement of the rotor along X, Y and Z axis is respectively x, y and z, and rotor is respectively around the angular displacement of X, Y-axis
α, β, the angular speed of rotor about the z axis are ω, and magnetic bearing acts on making a concerted effort and resultant moment respectively f in rotor geometric centerx、fy、
fzAnd Px、Py, rotor is respectively J around X, Y and Z axis rotary inertiax、JyAnd Jz。
(2) D'Alembert's principle is based on, rotor inertia axle is obtained and deviates interference of the unbalance mass, square to rotor caused by geometrical axis
Power.
If apex rotor is A ends, there is unbalance mass, square g in the duplicate removal face of A endsa=maR, wherein maFor imbalance in the duplicate removal face of A ends
Quality, r is duplicate removal radius.
Ignore external interference, following five equations can be listed according to D'Alembert's principle：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mrow>
<mi>g</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mrow>
<msub>
<mi>ag</mi>
<mi>a</mi>
</msub>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mrow>
<mi>g</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mrow>
<msub>
<mi>ag</mi>
<mi>a</mi>
</msub>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>z</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>F</mi>
<mrow>
<msub>
<mi>ag</mi>
<mi>a</mi>
</msub>
<mi>y</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>l</mi>
<mrow>
<mi>a</mi>
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>F</mi>
<mrow>
<msub>
<mi>ag</mi>
<mi>a</mi>
</msub>
<mi>x</mi>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>l</mi>
<mrow>
<mi>a</mi>
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
Wherein Fax、Fay、FazComponent of the perturbed force in X, Y, Z axis suffered by rotor A end is represented respectively,
Fgx、FgyRepresent rotor inertial forces in X, the component of Y-axis, F respectivelyagax、FagayRepresent that rotor unbalance moment of mass draws respectively
The perturbed force risen is in the component in X, Y-axis, FlRepresent active force of the deflection magnetic bearing on flywheel rotor, Mgx、MgyRepresent respectively
Inertia force represents that deflection magnetic bearing coil installs radius to being consolidated in the inertia couple of X-axis and Y-axis in flywheel equatorial plane, R,
lanRepresent rotor A end duplicate removal face to the distance of rotor center plane.
Because flywheel rotor is that inertia force is zero, i.e. F on inertia axially symmetric structuregx=Fgy=0, so unbalance mass, square
It is to the perturbed force of rotor：
<mrow>
<msub>
<mi>F</mi>
<mi>a</mi>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>a</mi>
<mi>z</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mo>-</mo>
<mfrac>
<mrow>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>y</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mrow>
<mi>a</mi>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mfrac>
<mrow>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>x</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mrow>
<mi>a</mi>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
If rotor bottom is B ends, similarly unbalance mass, square in B ends is gb=mbR, wherein mbFor imbalance in the duplicate removal face of rotor B ends
Quality, r is duplicate removal radius, then rotor B ends unbalance mass, square is to the perturbed force of rotor：
<mrow>
<msub>
<mi>F</mi>
<mi>b</mi>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>b</mi>
<mi>x</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>b</mi>
<mi>y</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>b</mi>
<mi>z</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mo>-</mo>
<mfrac>
<mrow>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>y</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mrow>
<mi>b</mi>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mfrac>
<mrow>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>+</mo>
<msub>
<mi>M</mi>
<mrow>
<mi>g</mi>
<mi>x</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>l</mi>
<mrow>
<mi>b</mi>
<mi>n</mi>
</mrow>
</msub>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, Fbx、Fby、FbzComponent of the perturbed force in X, Y, Z axis suffered by rotor B ends, l are represented respectivelybnRepresent rotor B ends
Distance of the duplicate removal face to rotor center plane.
(3) in order to ensure the uniformity of magnetic suspension spherical flywheel stator air gap, it is considered to caused by rotor centroid offset geometry axle partially
The heart is away from obtain eccentric throw is to the perturbed force of rotor：
<mrow>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>x</mi>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>y</mi>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>z</mi>
<mn>1</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<msup>
<mi>m&omega;</mi>
<mn>2</mn>
</msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mo>-</mo>
<msub>
<mi>e</mi>
<mi>m</mi>
</msub>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&rho;</mi>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mo>(</mo>
<mi>&omega;</mi>
<mi>t</mi>
<mo>+</mo>
<mi>&phi;</mi>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>-</mo>
<msub>
<mi>e</mi>
<mi>m</mi>
</msub>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&rho;</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mo>(</mo>
<mi>&omega;</mi>
<mi>t</mi>
<mo>+</mo>
<mi>&phi;</mi>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>e</mi>
<mi>m</mi>
</msub>
<mi>sin</mi>
<mi>&rho;</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein Fux1、Fuy1、Fuz1Represent component of the eccentric throw to the perturbed force of rotor in X, Y, Z axis, emRepresent flywheel rotor
Eccentric throw, Φ is the phase of eccentric throw, and ρ is eccentric throw and magnetic bearing central plane angle.
(4) consider that suspending power crosses the rotor centre of form and deflection negative moment that only barycenter is caused, and obtain thus caused perturbed force
For：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>x</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>y</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
<mo>-</mo>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>z</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>e</mi>
<mi>m</mi>
</msub>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&rho;</mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
I.e.：
<mrow>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>x</mi>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>y</mi>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>F</mi>
<mrow>
<mi>u</mi>
<mi>z</mi>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>F</mi>
<mi>l</mi>
</msub>
<mo>&CenterDot;</mo>
<mn>2</mn>
<mi>R</mi>
</mrow>
<mrow>
<msub>
<mi>e</mi>
<mi>m</mi>
</msub>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mi>&rho;</mi>
</mrow>
</mfrac>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein Fux2、Fuy2、Fuz2Represent component of the deflection negative moment to the perturbed force of rotor in X, Y, Z axis.
(5) above-mentioned three kinds of perturbed forces are considered, the rotor displacement amount under perturbed force is acted on is changed respectively by transition matrix
For the displacement under the displacement under sensor coordinate system and magnetic bearing coordinate system, and two kinds of displacements are applied to control respectively
In device and magnetic bearing, three kinds of perturbed forces are suppressed using feedforward suppressing method.
Rotor-magnetic bearing transition matrix is E, and the vertical range of sensor to XOY plane is la, level of the sensor to XOZ planes
Distance is l, then rotor-sensor transition matrix TsFor：
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mfrac>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<msqrt>
<mrow>
<msup>
<mi>l</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
</mtd>
<mtd>
<mfrac>
<mi>l</mi>
<msqrt>
<mrow>
<msup>
<mi>l</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mfrac>
<mi>l</mi>
<msqrt>
<mrow>
<msup>
<mi>l</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
</mrow>
</mtd>
<mtd>
<mfrac>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<msqrt>
<mrow>
<msup>
<mi>l</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>l</mi>
<mi>a</mi>
</msub>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mfrac>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
The numerical value of disturbance torque is obtained by calculating, feedforward compensation is carried out to disturbance torque by controller, is realized uneven to rotor
The suppression of weighing apparatus vibration.Condition, which is fully compensated, according to feedforward control to obtain：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>f</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mo>+</mo>
<mi>E</mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>u</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mo>+</mo>
<mi>E</mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>p</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mo>+</mo>
<mi>E</mi>
<mo>=</mo>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
In formula, Gf(s)、Gu(s)、Gp(s) unbalance mass, square, eccentric throw are represented respectively and deflect the feedforward transmission of negative moment
Matrix, Gw(s) power amplifier is represented.It can be obtained according to above formula：
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>f</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<mo>=</mo>
<mo>-</mo>
<msup>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>f</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<mo>=</mo>
<mo>-</mo>
<msup>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>G</mi>
<mi>p</mi>
</msub>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
<mo>=</mo>
<mo>-</mo>
<msup>
<msub>
<mi>k</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<msub>
<mi>G</mi>
<mi>w</mi>
</msub>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mtd>
</mtr>
</mtable>
</mfenced>
Above formula belongs to stable state and compensated entirely, is easily realized in control.
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| CN108909524A (en) * | 2018-03-16 | 2018-11-30 | 江苏大学 | Magnetic suspending flying wheel battery zero-power decoupling controller for electric vehicle based on dynamic model |
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